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Generalized Tate Cohomology
J. P. C. Greenlees and J. P. May
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Memoirs of the American Mathematical Society
1995; 178 pp; softcover
Volume: 113
ISBN-10: 0-8218-2603-4
ISBN-13: 978-0-8218-2603-4
List Price: US$47
Individual Members: US$28.20
Institutional Members: US$37.60
Order Code: MEMO/113/543
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This book presents a systematic study of a new equivariant cohomology theory \(t(k_G)^*\) constructed from any given equivariant cohomology theory \(k^*_G\), where \(G\) is a compact Lie group. Special cases include Tate-Swan cohomology when \(G\) is finite and a version of cyclic cohomology when \(G = S^1\). The groups \(t(k_G)^*(X)\) are obtained by suitably splicing the \(k\)-homology with the \(k\)-cohomology of the Borel construction \(EG\times _G X\), where \(k^*\) is the nonequivariant cohomology theory that underlies \(k^*_G\). The new theories play a central role in relating equivariant algebraic topology with current areas of interest in nonequivariant algebraic topology. Their study is essential to a full understanding of such "completion theorems" as the Atiyah-Segal completion theorem in \(K\)-theory and the Segal conjecture in cohomotopy. When \(G\) is finite, the Tate theory associated to equivariant \(K\)-theory is calculated completely, and the Tate theory associated to equivariant cohomotopy is shown to encode a mysterious web of connections between the Tate cohomology of finite groups and the stable homotopy groups of spheres.

Readership

Research mathematicians.

Table of Contents

  • Part I: General theory
  • Part II: Eilenberg-Maclane \(G\)-spectra and the spectral sequences
  • Part III: Specializations and calculations
  • Part IV: The generalization to families
  • Appendix A: Splittings of rational \(G\)-spectra for a finite group \(G\)
  • Appendix B: Generalized Atiyah-Hirzebruch spectral sequences
  • Bibliography
  • Index
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